Hjelmslev's theorem

In hyperbolic geometry, any two triangles with the same three angles are congruent. Their angles determine their side lengths, unlike in Euclidean geometry.

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Hyperbolic triangle: A triangle formed by three geodesics in hyperbolic geometry. The theorem applies to these triangles, whose angle data determine their congruence.
Euclidean geometry: Geometry based on Euclid’s postulates, including the parallel postulate. Euclidean triangles with the same angles can be differently sized and noncongruent.
Congruence (geometry): A relation between figures that can be mapped onto one another by distance-preserving transformations. The theorem asserts this stronger relation, not merely matching shape.
Johannes Hjelmslev: A Danish mathematician known for work in geometry and the foundations of mathematics. His name is attached to the hyperbolic triangle congruence theorem.
Hyperbolic law of cosines: A relation connecting the sides and angles of a triangle in hyperbolic geometry. It translates shared angles and side relations into the same hyperbolic triangle geometry.
Similarity (geometry): The study of figures related by transformations that preserve shape while allowing scale to change. In Euclidean geometry, equal angles establish triangle similarity, not necessarily congruence.
Hyperbolic distance: The distance between points measured by the hyperbolic metric. Congruence concerns preservation of these distances, including the triangle’s side lengths.
Nikolai Lobachevsky: A Russian mathematician who independently developed hyperbolic geometry in the nineteenth century. His work established the geometric setting in which the theorem makes sense.
Gauss–Bonnet theorem: A theorem relating a surface’s curvature to the geometry and topology of regions on it. For hyperbolic triangles, it makes angle defect proportional to area.
AAA similarity: The Euclidean triangle criterion that equal corresponding angles imply similarity. Its conclusion stops at similarity, whereas Hjelmslev’s theorem gives hyperbolic congruence.